Wetting dynamics and contact angles of powders studied through capillary rise experiments

Colloids and Surfaces A: Physicochem. Eng. Aspects 436 (2013) 371–379

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Wetting dynamics and contact angles of powders studied through capillary rise experiments Andrea Depalo, Andrea C. Santomaso ∗ APTLab – Advanced Particle Technology Laboratory, Università di Padova, Dipartimento di Ingegneria Industriale, via Marzolo 9, 35131 Padova, Italy

h i g h l i g h t s

g r a p h i c a l

a b s t r a c t

• A method to measure advancing and equilibrium contact angles in powders is proposed. • A new equivalent capillary radius for packed bed is used. • We improve the measure of powder effective porosity accounting for their flowability.

a r t i c l e

i n f o

Article history: Received 25 March 2013 Received in revised form 5 June 2013 Accepted 25 June 2013 Available online xxx Keywords: Powder wettability Capillary rise Contact angle Capillary pressure

a b s t r a c t Wettability is an important property involved in the industrial use of granular solids and powders. It is commonly described with the contact angle and an experimental method for its determination in dynamic conditions is proposed in this work. The method is based on the capillary rise of the wetting liquid into a packed bed of the material under analysis. Differently from the classical Washburn method, the packed bed is closed to the atmosphere and the air pressure increase is measured allowing to evaluate the powder contact angle through a dynamic balance of the pressure forces. In the expression of such forces a new equivalent capillary radius for the powder bed is used based on an alternative definition of the particle equivalent diameter. This diameter is closely related to the length of the three phase line which divide the wet portion of the bed from the dry one and mirrors the physics of wetting process better than the classical Sauter diameter. A way to determine it with optical microscopy is given. Also the measure of the packed bed porosity (entering in the equivalent capillary radius definition) has been improved by using the effective porosity concept [Hapgood et al., J. Coll. Interface Sci. 253 (2002) 353–366] and by modifying the way of estimating it. The proposed experimental technique, coupled to the theoretical model for the packed bed, can describe accurately the packed bed geometry and the wetting dynamics by following the changes of the contact angle from its initial maximum value up to the final equilibrium one. © 2013 Elsevier B.V. All rights reserved.

1. Introduction The wettability is a physicochemical property which has an important role in the industrial use of powdered and granular solids. Granulation, flotation and dissolution are just some examples of the several operations where powder wettability is important. In the wet granulation process the size of primary

∗ Corresponding author. Tel.: +39 049 8275491; fax: +39 049 8275460. E-mail address: [email protected] (A.C. Santomaso). 0927-7757/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.colsurfa.2013.06.040

particles is increased by binding them into agglomerates using capillary forces [1,2] and the affinity between the liquid binder and the solid impact on the final size of the granules [3]. In the mineral processing, flotation is widely used to quickly and efficiently separate valuable minerals from the gangue minerals. The separation is controlled to a large extent by the relative wettability of mineral particles in a pulp [4,5]. In the dissolution process of commercial powders (for instance instant drink powders) a larger affinity between the liquid and the solid phase leads to a faster wetting and sinking of the particles and therefore increases the dissolution rate [6]. In all these operations the contact angle plays a major

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Nomenclature b dOM dpa d(x) dsv g h Hsh K l P0 Pf R Req Rh t x  LV  SL  SV P Pcap ε εeff εtap εmin   A E R b L M s 

immersion depth of the sample holder in the liquid [m] diameter obtained from optical microscopy [m] mean equivalent diameter related to the ratio (projected area)/(wet perimeter) [m] diameter of the exposed circumference in a spherical particle cross sectioned at height x [m] Sauter mean diameter [m] gravitational acceleration [m/s2 ] height reached by the liquid [m] height of the sample holder assumed cylindrical with constant section [m] proportionality factor between height and pressure [m/Pa] length covered by the liquid in the bed [m] atmospheric pressure [Pa] final pressure after the compression [Pa] radius of the cylindrical capillary tube [m] equivalent capillary radius [␮m] hydraulic radius [m] time [s] (for a particle) height covered by the liquid [m] liquid–vapor surface tension [J/m2 ] solid–liquid surface tension [J/m2 ] solid–vapor surface tension [J/m2 ] relative pressure (referred to the atmospheric one) [Pa] capillary pressure [Pa] generic porosity of the powder bed [–] effective porosity [–] tap porosity [–] porosity value related to the maximum packing of the powder bed [–] viscosity of the wetting liquid [Pa s] generic value of the contact angle [◦ ] advancing contact angle [◦ ] equilibrium contact angle [◦ ] receding contact angle [◦ ] bulk density of the powder bed [kg/m3 ] density of the wetting liquid [kg/m3 ] density value related to the maximum packing of the powder bed [kg/m3 ] solid intrinsic density [kg/m3 ] tortuosity factor [–]

role in the wetting process. The contact angle is an indicator of the liquid–solid affinity and originates from the equilibrium balance between adhesive and cohesive forces [7]. The Young equation: cos E =

SE − SL LV

(1)

links the contact angle to the three surface tensions involved in the formation of a sessile drop of liquid on a flat horizontal solid surface in presence of a third phase (a gas or an immiscible liquid), as represented in Fig. 1a. The three surface tensions are the solid–liquid  SL , the solid–vapor ␥SV and the liquid–vapor  LV tension. The condition of zero contact angle is referred to as perfect wetting while the cases with angles lower or greater than 90◦ are referred to as wetting and non-wetting, respectively. In the above definitions the drop is in static equilibrium and therefore the angle is called equilibrium contact angle  E . In several industrial application however the

liquid is in motion with respect to the solid so that also an advancing  A and a receding  R contact angle can be defined (Fig. 1b) [8]. The evaluation of the contact angle through Eq. (1) is not a trivial task because the solid–vapour and the solid–liquid tensions are difficult to evaluate [7] so that the direct measurement is the preferred method in practice. Considering powders however, even the direct measurement is not trivial at all. Because of the discrete nature of the particles, the surface of powder sample is irregular and porous. The droplet of liquid can spread and sink into the porosities with a kinetics that can be also very fast, depending on the solid–liquid affinity, on the particle size, on the consolidation state of the powder and obviously on the viscosity of the liquid. As an alternative to the sessile drop method the contact angle of powders can be evaluated with the rising liquid method [7]. It relates the contact angle to the rising rate of the liquid into a packed bed of the powder under test. The packed bed is assumed to be equivalent to a bundle of capillary tubes with circular section and equivalent (or effective) radius, Req . The driving force is assumed to result only from the capillary pressure (the hydrostatic head contribution related to the height h reached by the liquid is neglected). A proportionality between the square of the height h reached by the liquid in the bed and the time t can be found. This relationship is expressed in the Washburn’s equation (2) as: h2 =

Req LV cos A t 2

(2)

where  is the viscosity of the wetting liquid. As alternatives to the direct measurement of the height h, also the weight [9] or the overpressure (generated by the rising liquid when the sample holder is closed to the atmosphere) can be measured [10,11]. Unfortunately in Eq. (2) besides  A also Req is unknown and needs to be evaluated. Req can be determined by using a reference liquid assumed to “perfectly” wet the powder under test. For a perfectly wetting liquid the cos  A ∼ 1 so that Req can be found from the experimental evaluation of the h2 (t) slope. Provided that Req is independent of the wetting liquid and does not change during the packing procedure, it can be used then in Eq. (2) to calculate cos  A by measuring the slope of h2 (t) for the liquid under test. The evaluation of Req is in any case a critical step in the determination of the contact angle. The use of a perfectly wetting liquid can present some drawbacks. First of all it is necessary to find a liquid that perfectly wets the powder under study [7,12]. This can be problematic when several materials have to be analyzed so that a different perfectly wetting liquid could be necessary for each of them [12]. It is common to assume as perfectly wetting liquid, the most wetting one among those available to whom is doing the experiments [6] but this produces results that depend on the choice of the reference liquid. Moreover the method requires that the geometry of the powder bed (i.e. Req ) is the same in the tests with the reference liquid and with the liquid under study. This is possible only applying a systematic procedure that guarantees a reproducible packing of the powder among different experiments [13]. A different approach is that of modeling, under simplifying assumptions, the structure of the solid bed and the solid–liquid physical interaction during wetting [13,14] so that to find an analytical expression for Req . This approach has been followed in the present work. In particular, with respect to published literature, an alternative particle equivalent diameter and an improved way of estimating the effective porosity of the packed bed have been proposed for the calculation of Req . The overpressure generated by the liquid rise has been measured in order to estimate the contact angles as in [10,11], but differently from Iveson et al. [10] the perfectly wetting liquid was not necessary (because of the analytical estimation of Req ) and differently from Wei et al. [11] an

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Fig. 1. Sketches of the equilibrium contact angle involved in the formation of a sessile drop on a horizontal flat solid surface (a) and of the advancing and receding angles typical of non-equilibrium conditions (b).

integration of the pressure balance equation over short time period was proposed in order to improve the estimation of cos . In Section 2 the experimental set-up and the details for the sample preparation are given. Then in Section 3 the physical model for the liquid rising is presented with details on the numerical integration of the pressure balance equation. Also the quantities necessary to define the analytical expression of Req are given here. In particular a new equivalent diameter and an improved way of estimating the effective porosity of the packed bed are proposed. Section 4 finally presents the experimental results and comments on the dynamic and equilibrium contact angles obtained with the proposed method. 2. Materials and methods 2.1. The experimental set-up The experimental set-up, shown in Fig. 2, consisted of a sample holder where the powder was allocated. The sample holder was a cylindrical HDPE container (height = 63.4 mm, internal diameter = 12.5 mm). The top was closed and fitted to a silicone tube connected to a differential pressure sensor. The bottom instead was closed with a removable permeable surface. When opened, it allowed to fill the cylinder with the powder and once closed it retained the powder while permitting the inflow of the wetting liquid. The permeable surface was obtained by coupling a cotton cloth with a filter paper disk placed between the cotton cloth and the powder to avoid the escape of particles through the cloth fibers. The capillary rise started when a laboratory plate with the liquid was put in contact with the bottom of the sample holder. The rise caused the compression of the air in the sample holder and the overpressure variation was detected by a differential pressure sensor (Honeywell 164PC01D37, pressure range 0–10 in H2 O) working with a stabilized power supply (STAB mod. AR25 Output 0–30 V to 2.5 A) at 8 V DC. The pressure sensor and the sample holder were connected with a three-way connection to an air buffer reservoir consisting of a thermally insulated plastic bottle (300 ml volume). Thermal insulation was required to protect the overpressure signal from the ambient temperature fluctuations. The air reservoir contributed to damp down the pressure signal fluctuations and to reduce an undesired phenomenon occurring in some experiments. This phenomenon, observed also by Iveson et al. [9], consisted in the escape of air bubbles from the bottom of the sample holder which biased the overpressure signal. It occurred in some tests, in particular with monohydrate lactose and microcrystalline cellulose, typically at the beginning of the capillary rise. It was due to the initial fast overpressure increase which forced part of the air to exit from the bottom. The use of a larger air volume reduced the overpressure and the bubbling was drastically attenuated or even eliminated. The tension signal coming from the pressure sensor and proportional to the overpressure value, was acquired by a data logger (Measurement Computing, USB-1208 FS) and visualized and saved through the software TracerDAQ. A preliminary calibration with a water manometer was adopted in order to convert the original digital signal in volt into the corresponding pressure signal. The

pressure signals were filtered to reduce noise. All the data manipulations and analyses were implemented with the software Matlab (MATLAB 7.5, The MathWorks, Inc., Natick, Massachusetts, United States). 2.2. Sample preparation and experimental procedure The bed of powder was packed at the bulk density, b , by imposing a predetermined number of vertical tap to the sample holder. Details of the apparatus used to this scope are given elsewhere [15]. In general b was measured after application of 100 taps to the sample holder. Only for monohydrate lactose and calcium carbonate B, a reduced number of taps (20) was applied in order to limit the formation of cracks in the material. Cracks formation was related to the contraction of the packing during the wetting process. After tapping, the sample holder was placed on the support frame and connected to the differential pressure sensor and to the air reservoir. The experiment started by measuring the atmospheric pressure as the base line for the signal. After 60 s the bottom of the sample holder was immersed in the liquid by lifting the liquid reservoir through the lift table. The increase of the overpressure was recorded by the data logger. Because of the tapping procedure repeatable values of the bulk density and of the air overpressure trends were obtained (each test was repeated three times). At the end of the experiment the sample holder was removed from the system and the height of the risen liquid measured by visual inspection. 2.3. Granular materials and wetting liquids The granular materials used in this work were: glass ballottini, river sand, calcium carbonate from quarry (indicated as type A), pharmaceutical calcium carbonate USP (indicated as type B), monohydrate lactose, microcrystalline cellulose Avicel PH101 (MCC) and ground coffee powder. They are listed in Table 1 and classified according to their flow propensity as cohesive or free-flowing materials. Deionized water was always used as wetting liquid except with the monohydrate lactose for which a saturated solution of lactose in water was used (see Table 1). This was necessary to avoid the solubilization of the lactose powder during the rise experiment. Table 2 reports some physical properties (density L , surface tension  LV and viscosity ) of the wetting liquids.

Table 1 Materials used, flow propensity and wetting liquids. Material

Flow propensity

Wetting liquid

Glass ballottini River sand Calcium carbonate A Calcium carbonate B Monohydrate lactose MCC Avicel PH101 Coffee powder

Free-flowing Free-flowing Free-flowing Cohesive Cohesive Cohesive Cohesive

Deionized water Deionized water Deionized water Deionized water Lactose saturated solution Deionized water Deionized water

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Fig. 2. Sketch of the experimental set-up. The bottom of the sample holder is immersed in the liquid while the top is connected to a differential pressure sensor in order to measure the overpressure increase due to the rising liquid.

Table 2 Properties of the liquids used.

Eq. (5).

Wetting liquid

Deionized water

Lactose saturated solution*

Density [kg/m3 ] Surface tension [J/m2 ] Viscosity [Pa s]

1000 7.28 × 10−2 1.00 × 10−3

1071 7.16 × 10−2 1.10 × 10−3

*

cos  =

 2LV

R (t2 − t1 )

  4 2 K 2



The rise of the liquid in the packed bed caused the compression of the air in the sample holder and continued until the pressures forces were perfectly balanced. The dynamics of the liquid is described by



2LV cos  + L gb R

− ( P + L gh)

(3)

In this equation, l is the length covered by the liquid in the bed, b is the immersion depth of the sample holder in the liquid and P is the relative overpressure reached by the air in sample holder. On the LHS the viscous resistance to the flow is expressed by the Poiseuille law valid for a Newtonian liquid in laminar flow. The tortuosity has been introduced to take into account the difference between the height h and the length covered by the rising liquid.  was assumed equal to 25/12 [14]. Due to the small size of the pores the inertial resistances are negligible [11] and therefore do not appear in Eq. (3). The RHS instead contains four terms which contribute to the overall pressure difference at the extremities of the powder bed. The first term is the capillary pressure (the Laplace equation), the second is the liquid head due to the sample holder immersion depth, b, the third is the measured overpressure and the last is the hydrostatic head related to the height, h, covered by the rising liquid. According to Wei et al. [11], the height of the liquid that rises in the powder bed is proportional to the air pressure increase in the sample holder as in Eq. (4), where K is a constant (it will be verified in the following that K is not constant but can be assumed constant). h = K P



[ P(t2 )2 − P(t1 )2 ]



Pdt − L gb(t2 − t1 )

(5)

t1

3. The integration of the pressure forces balance



t2

+(1 + L gk)

Values taken from Hapgood et al. [16].

8 2 h dh = R2 dt

R2

(4)

By replacing h with K P, Eq. (3) become homogeneous in P and after some algebraic manipulation can be integrated in the generic time interval t2 − t1 obtaining the explicit expression for cos ,

Here, differently from Wei et al. [11] the determination of cos  was based on the numerical integration of the pressure profile instead of using the differential equation as it is. In doing so the method resulted much less sensitive to the noise of the pressure sensor signal, which was observed to strongly affect the final evaluation of cos  with the Wei et al. method. In order to obtain an accurate numerical integration of Eq. (5) short time intervals, t2 − t1 , were taken along the P(t) profile. In these time intervals both K and  were considered constant with a negligible error. Time intervals of 10 s were considered optimal and were adopted in all the experiments, allowing to obtain the trend of  at discretized time values. The rising velocity of the liquid was maximum at the beginning of the experiment and decreased with time because of the increasing overpressure and hydrostatic head in the sample holder. A verification of the laminarity of the flow, at the very beginning of the experiment, was therefore necessary. It was found that the Reynolds number never exceeded the order of magnitude of 10−3 which largely satisfied the laminarity condition for the flow in packed beds (Re < 10) [13]. As a consequence of the dynamic nature of the overpressure measurements, the contact angle estimated with this method was not constant but decreased going from a maximum initial value to an asymptotic final value at equilibrium. All the angles measured with the liquid in motion could be therefore considered as advancing contact angles,  A , while the asymptotic one as the equilibrium contact angle,  E . In order to evaluate the contact angle with Eq. (5) however, besides the measured trend for P(t) also the constant K, the equivalent capillary radius Req and the porosity of the powder bed, ε, must be known. 3.1. Determination of K Let us first consider a simplified system made of an empty cylinder of constant cross section S and length Hsh with a piston giving an initial pressure P0 equal to the atmospheric one. After pushing

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real packed bed. While it is clear that the specific surface area is a relevant properties for the flow of a fluid through a packed bed of particles (it is indeed related to the drag of the fluid on the whole particle surface), it is not clear why it should be relevant also in capillary phenomena where the driving force is related to the length of the three phase line which divide the wet portion of the bed from the dry one according to the model sketched in Fig. 4. A different equivalent diameter of the particles should be therefore considered in the definition of Req . Req can be defined starting from the classical approach which uses the hydraulic radius, Rh [13,16]: cross section available for flow wetter perimeter

Rh =

Fig. 3. Sketch of a compressed air volume in a cylinder for modeling K.

(8)

where the cross section available for flow and the wet perimeter are identified by the cutting plane in Fig. 4b. Both terms can be divided by the cross sectional area of the sample holder and the equation can be rewritten as in Eq. (9) due to the fact that for a random packing of particles the sectional void fraction coincides to the volumetric void fraction (i.e. to the porosity ε) (see Appendix 1 or [18]). Rh = Fig. 4. (a) Packed bed of spheres wetted up to a certain height h and (b) sketch of the sectional area at height h between the wet and the dry portion. The capillary rise scales with the length of the three phase line which divide the wet portion of bed from the dry.

the piston to a height h, the pressure increases to a value Pf (Fig. 3). For constant temperature the Boyle law gives: P0 SHsh = Pf S(Hsh − h)

(6)

overall cross section of the particles

=

ε 1−ε

(9)

The ratio between the overall cross section of the particles and their overall perimeter in Eq. (9) can be expressed in terms of the equivalent diameter of the sphere that has the same value of this ratio. This average diameter can be indicated as dpa (where p stays for perimeter and a for projected area):

 d2 pa

area of the projected sphere = perimeter of the projected sphere

4

dpa

=

dpa 4

(10)

If the powder bed is assumed as a bundle of cylindrical capillary tubes with radius Req , this is related to Rh according to:

and after some algebraic manipulation h Hsh Hsh = =K = (P0 + P) (101325 + P) P

 overall perimeter of the particles 

(7)

where P is the relative pressure (Pf − P0 ). Eq. (7) can be used also in a real experiment (cylinder filled with the powder) where Hsh has to be interpreted as a fictitious length related to the volume available to the air in the sample holder considered as it was an empty cylinder of constant cross section. At the end of each test, the overpressure P (by sensor data) and the height of the liquid h (by visual inspection) were measured. According to Eq. (7) it was therefore possible to obtain K and Hsh at the end of the each experiment. However since Hsh is a constant it was possible to verify, through Eq. (7), that K varied very little during the experiments (see Fig. 7 in Section 4) so that assuming K as a constant in the integration intervals was a reasonable approximation.

Rh =

2

Req

2 Req

=

Req 2

(11)

Considering Eqs. (9)–(11) the equivalent capillary radius can be finally defined as a function of dpa : Req =

dpa ε 2 1−ε

(12)

The dpa is the equivalent average spherical diameter representative of the irregular particles sketched in Fig. 4b and analogously to dsv it has to be estimated from independent experimental measurements. It can be obtained for example from optical microscopy measurements of the particles under study. For the details on the estimation of dpa from optical microscopy measurements and on the related issues see Appendix 2.

3.2. Determination of the equivalent capillary radius

3.3. Determination of the porosity

The driving forces in the capillary rise are the adhesion forces between the solid and the liquid phases. They act at the boundaries between the wet portion of the powder bed and the dry one. Let us therefore consider a packed bed of spherical particles wet up to a certain height h by a rising liquid. This height ideally identify a transversal cutting plane (Fig. 4a) which separates the wet portion from the dry one and identify the boundaries of the cut particles (the three phase line) where the capillary forces act (Fig. 4b). The typical literature approach uses the Sauter mean diameter as the equivalent average diameter for describing the packed bed [14,17]. The Sauter mean diameter (or dsv ) is the size of the sphere which gives a packing with the same specific surface area as the

The porosity appears in the expression of Req and is the ratio between the void volume in the packed bed and the total volume. It is related to the solid density S and to the bulk density b of the powder according to Eq. (13): ε=1−

b s

(13)

However not all the voids which contribute to ε give a contribution to the capillary rise phenomenon. Some of the larger pores indeed can cause the stop of the capillary rise so that not all of the total porosity is available for the liquid. This fact is well known in the literature [16,19] and can be taken into account in different ways. Following the approach by Hapgood et al. [16] an effective porosity

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Fig. 5. Overpressure trends obtained in the three repeated experiments performed for river sand.

Fig. 6. Air pressure increase P trend in two experiments with glass ballottini and calcium carbonate B.

εeff related to the pores that actually contribute to the capillarity can be defined. Hapgood et al. [16] propose the following equation:

εeff = εtap 1 − ε + εtap



(14)

where εtap is the tap porosity obtained after the imposition of a tapping action or vibration to the bed of powder. The tap density, in the authors intention, should be the maximum bulk density that can be achieved by the powder bed. However, Santomaso et al. [15] showed that the procedure to achieve the maximum bulk density in a packed bed depends on the flowability of the powder. For powders behaving as cohesive the maximum density can be obtained by a tapping action (800 taps were used in this work), meanwhile for free-flowing powders higher bulk densities can be obtained by particle pluviation. It consists in filling a 25 cc container by dispersing the material through a metallic grid (mesh size 4 mm) positioned at an height of 400 mm over the container [16]. The resulting bulk density is called dispersed density and is calculated as usual as the ratio between the weight of the packed powder and the container volume. Accordingly, the porosity εtap in Eq. (14) was replaced by a generic minimum porosity (εmin ) corresponding to the maximum bulk density obtained by tapping or pluviation depending on powder flowability. A rough indication of powder flowability is given in Table 1. The solid density, s , appearing in Eq. (13) was measured by picnometry using distilled water as picnometric liquid for all the powders, excepted for the monohydrate lactose for which ethanol 99 (w/w%) was used because of lactose water solubility. 4. Results and discussion The physical properties of the particles and of the packed bed, necessary for the determination of the contact angle, are collected in Table 3. In particular the solid density and the bulk density have been measured as described in Section 2 while the bulk porosity, the maximum density, the effective porosity, the mean diameter and the equivalent capillary radius have been calculated according to the methods described in Section 3. Also the overpressure due to the capillary rise was measured and Fig. 5 shows three typical overpressure profiles. As expected the slope of the profiles progressively decreased with time due to the fact that the system approached to the equilibrium condition. It can also be observed that the repeatability of the three experiments was quite good, mainly because of the systematic packing procedure adopted. The profiles in Fig. 5 refer to the river sand but similar trends were observed for the other materials. The main difference was in the rate of the pressure increase which was peculiar to each material. In Fig. 6 for example the overpressure increase for glass ballottini and for calcium carbonate B are compared. The different overpressure increase rate was related mainly to the value of the equivalent radius Req (Table 3). The materials with larger Req ,

Fig. 7. Proportionality factor K trend relative to two experiments with glass ballottini and calcium carbonate B.

showed a dynamics much faster than that typical of smaller Req (monohydrate lactose, MCC and calcium carbonate B). Also K was evaluated as a function of time in order to verify that it varied little during the experiments. K was calculated at each time step using Eq. (7) and Fig. 7 shows the typical trends for K (same materials of Fig. 6). They confirm that K can be considered constant in the short time intervals (10 s) used for integration of Eq. (5). The variation of K in such time intervals never exceeded the 0.5%. With the overpressure profiles, K and Req , at hand the contact angle was obtained at each time step by integrating the pressure forces balance Eq. (5). In Fig. 8 the contact angles for glass ballottini and calcium carbonate B are shown as a function of time. For glass ballottini the contact angle was characterized by a rapid approach to the equilibrium value and after 600 s the overpressure was very close to the asymptote. In this case the extrapolation of the final equilibrium value was straightforward. For calcium carbonate B instead the pressure at the end of the experiment was still far from the equilibrium and the extrapolated value for the equilibrium angle has to

Fig. 8. Contact angle values obtained for glass ballottini and calcium carbonate B. In the above experiments all the contact angle values should be defined as advancing contact angles since the liquid was in motion during the measurement.

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Table 3 Physical properties of the packed bed and of the particles used in the experiments. Materials

Solid density S [kg/m3 ]

Bulk density b [kg/m3 ]

Bulk porosity ε [–]

Maximum bulk density M [kg/m3 ]

Effective porosity εeff [–]

Mean diameter dpa [␮m]

Equivalent capillary radius Req [␮m]

Glass ballottini River sand Calcium carbonate A Calcium carbonate B Monohydrate lactose MCC Avicel PH101 Coffee powder

2574 2626 2700 2930 1455 1571 1068

1502 1462 1433 478 467 328 356

0.417 0.443 0.469 0.837 0.679 0.792 0.666

1586 1605 1557 582 625 360 362

0.371 0.368 0.404 0.773 0.508 0.755 0.657

195.2 303.1 528.8 6.0 15.7 12.9 188.4

57.7 88.2 179.2 10.3 8.1 19.9 180.4

Table 4 Advancing and equilibrium contact angles values obtained for all the materials analyzed. Material

 A [◦ ]

Glass ballottini River sand Calcium carbonate A Calcium carbonate B Lactose monohydrate MCC Avicel PH101 Coffee powder

82.9 69.0 72.0 88.6 88.2 81.0 80.8

*

 E [◦ ]* ± ± ± ± ± ± ±

0.9 0.2 0.5 0.3 0.7 1.3 0.4

71.4 25.1 33.7 77.8 81.7 58.0 42.1

± ± ± ± ± ± ±

 =  A −  E [◦ ] 0.2 5.5 2.8 3.5 4.2 3.4 3.2

11.5 43.9 38.3 10.8 6.5 23.0 38.7

Extrapolated values.

be taken with some caution. These different behaviors are related to the different air overpressure profiles shown in Fig. 6. In order to improve the estimation of the equilibrium value for the materials with small equivalent capillary radii, as calcium carbonate B, a longer sample holder should be used at the price of longer experiment duration, or an additional overpressure could be imposed in the sample holder at the price of a more complex experimental set-up. For this reason in Table 4 the initial maximum contact angle, obtained at the very beginning of the experiment when the pressure in the sample holder was close to the atmospheric one, is indicated as the advancing contact angle,  A . Table 4 also reports the estimation of the equilibrium contact angle obtained by the regression and the values of the difference between the advancing and the equilibrium contact angle. In some cases, for example for glass ballottini, the extrapolated equilibrium contact angle was unexpectedly large (71.4◦ ). The reason could be the presence of impurities on the surface of the material which was used as received (without any surface cleaning or conditioning). A verification of this value was possible using the sessile drop method on a monolayer of glass ballottini (Fig. 9). The measured value of  E was 66.6 ± 0.8◦ which was reasonably close to that obtained by capillary rise.

Fig. 9. Water droplet on a mono-layer of glass ballottini.

In some cases a large difference (up to ∼40◦ ) between the advancing and the equilibrium angle were observed ( in Table 4). This difference is the consequence of the deformation of the liquid meniscus in the capillary rise phenomena and essentially depends on the liquid velocity [7]. Even if the order of magnitude for  reported in Table 4 is in agreement with literature data for the case of the perfectly wetting liquids [7], a critical aspect of the results presented here remains the necessity of extrapolating the equilibrium contact angles (in particular for materials with small Req ). This is however a limit of the experimental set-up available for the tests and not a limit of the method. Using a longer sample holder (and/or imposing a counter pressure in the air reservoir) the system would reach the equilibrium faster allowing the direct measurement of  E also for packings with small Req . 5. Conclusions The aim of this work was the characterization of the wettability of different granular solids in dynamic conditions through the evaluation of their contact angles (advancing and equilibrium ones). An experimental method based on the study of the capillary rise of the wetting liquid in the powder under test was proposed. The sample holder with the powder was closed to the atmosphere and the overpressure increase (caused by the liquid rise) was related to wettability of the powder by using the integrated form of the balance of the pressure forces. In order to transform overpressure information into the contact angle values it was necessary to reconsider the model for the equivalent capillary radius and the way of measuring the packed bed porosity. The model for the equivalent capillary radius was related to the length of the three phase line which divide the wet portion of the bed from the dry one. This led to a redefinition of the equivalent particle diameter characterizing the bed of powder under test. A way for measuring this new equivalent spherical diameter from independent optical microscopy measurements was suggested. Also the packed bed porosity estimation was reconsidered by proposing a modified way of measuring the effective porosity. The results showed that by integrating the pressure forces balance over sufficiently short time periods (10 s) it was possible to follow accurately the evolution of the advancing contact angle and have an estimation of the equilibrium contact angle without the necessity of using a perfectly wetting liquid. The method showed good repeatability (as it can be inferred from the low the standard deviations), provided that samples were prepared following an accurate packing procedure. A critical point remained the estimation of the equilibrium contact angle (in particular for materials with small Req ) which was obtained by extrapolation from the advancing angle profile and suggestions to improve the measure have been presented in the paper.

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Considering the particles as spheres (Fig. A.1) the mean value dpa of the diameter d of the generic cross section can be defined as a function of x as



d(x) = 2

x(dOM − x)

where dOM is the maximum diameter of the silhouette obtained from optical microscopy. The average diameter can be obtained as:

 x=d dpa =

OMd(x)dx

x=0 x=d



x=0

OMdx

After normalization and change of variables, the integrand function and the integral become respectively: Fig. A.1. Sphere wetted up to an height x.

d∗ (x∗ )



d(x) = =2 dOM



Appendix 1. The following demonstration was first given by Gray [18] and shows that the 2-dimensional porosity (i.e. evaluated on a generic cutting plane) is equal to the 3-dimensional one for a randomly packed bed of particles. Let us consider a cube of a bed of particles of side L containing n particles of characteristic linear dimension d. Volume of the cube = L3 . Total volume of the particles = nkd3 , where k is a constant depending on shape. Bed porosity ε = 1 − kd3 /L3 . Two assumptions are made: (1) particles are randomly distributed in the bed and (2) the cube is cut by a plane parallel to one of its faces. The statistical probability of cutting any particular particle is d/L. The average area of each particle exposed as a result of cutting by the plane is approximately equal to the volume of the particle divided by its characteristic linear dimension: kd3 /d = kd2 Thus, the average area of the particles exposed as a result of cutting by the plane is: 2

3

kd d/L = kd /L The total cross sectional area is L2 so that the average free cross sectional area is: L2 − kd3 /L and the fraction of area available for flow is: (L2 − kd3 /L) =ε L2

Appendix 2. The analysis of the particles by optical microscopy gives the silhouette of the projected particles corresponding to the largest section of the particles. In a randomly packed bed however the area of each particle exposed as a result of cutting by the plane (Fig. 4b) can be in the range from zero to the largest section given by optical microscopy. The average exposed area (corresponding to the average diameter dpa ) must be evaluated.

dpa = dOM

x∗ =1

x∗ =0



x dOM



1−

 x  

d∗ (x∗ )dx∗

x∗ =1

dOM x∗ =1

= dOM dx∗



2



=2

x∗ (1 − x∗ ) .

x∗ (1 − x∗ )dx∗

x∗ =0

x∗ =0

The last integral is equal to /4 and therefore dpa =

dOM 4

The above integration corresponds also to the calculation of the average chord length of a circle. For the materials with smaller size (lactose, calcium carbonate B, MCC) an optical microscope (Olympus IX2-SP with magnification 20×) was used for the determination of dOM . For the materials with larger particle sizes (glass ballottini, river sand, calcium carbonate A and coffee powder) a scanner (Model CanoScan D1250 U2F) was used instead. The digital images were analyzed with the software UTHSCSA ImageTool 3.0 (developed at the University of Texas Health Science Center at San Antonio, Texas and freely available from the Internet). References [1] J. Lister, B. Ennis, The Science and the Engineering of the Granulation Processes, Kluwer Academic Publishers, Netherlands, 2004. [2] M. Cavinato, M. Bresciani, M. Machin, G. Bellazzi, P. Canu, A.C. Santomaso, Formulation design for optimal high-shear wet granulation using on-line torque measurements, Int. J. Pharm. 387 (1–2) (2010) 48–55. [3] M.E. Aulton, M. Banks, Influence of the Hydrophobicity of the Powder Mix on Fluidised Bed Granulation, in: Proceedings of the International Conference on Powder Technology in Pharmacy, Basel, Switzerland, Powder Advisory Centre, 1979. [4] C.A. Prestidge, J. Ralston, Contact angle studies of galena particles, J. Colloid Interface Sci. 172 (1995) 302–310. [5] T.T. Chau, A review of techniques for measurement of contact angles and their applicability on mineral surfaces, Miner. Eng. 22 (2009) 213–219. [6] L. Forny, A. Marabi, S. Palzer, Wetting, disintegration and dissolution of agglomerated water soluble powders, Powder Technol. 206 (2011) 72–78. [7] M. Lazghab, K. Saleh, I. Pezron, P. Guigon, L. Komunjer, Wettability assessment of finely divided solids, Powder Technol. 157 (2005) 79–91. [8] A. Siebold, M. Nardin, J. Schultz, A. Walliser, M. Oppliger, Effect of dynamic contact angle on capillary rise phenomena, Colloids Surf. A: Physicochem. Eng. Aspects 161 (2000) 81–87. [9] A. Siebold, A. Walliser, M. Nardin, M. Oppliger, J. Schultz, Capillary rise for thermodynamic characterization of solid particle surface, J. Colloid Interface Sci. 186 (1997) 60–70. [10] S.M. Iveson, S. Holt, S. Biggs, Contact angle measurements of iron ore powders, Colloids Surf. A: Physicochem. Eng. Aspects 166 (1999) 203–214. [11] Bigui Wei, Qing Chang, Caiyun Yan, Wettability determined by capillary rise with pressure increase and hydrostatic effects, J. Colloid Interface Sci. 376 (2012) 307–311. [12] L. Susana, F. Campaci, A.C. Santomaso, Wettability of mineral and metallic powders: applicability and limitations of sessile drop method and Washburn’s technique, Powder Technol. 226 (2012) 68–77.

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